Abstract
We study Boolean networks which are simple spatial models of the highly conserved Delta–Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained “patterns” that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.
Highlights
Lateral inhibition is a signalling mechanism that can induce the differentiation of cells in developing tissues (Sternberg 1993; Collier et al 1996)
For the asynchronous dynamics associated to the network F, we show that all the attractors found in the minimal trap space containing the state are reachable
In this work we gave some characterisations of the dynamics of simple Boolean models of the Delta–Notch system, complementing existing computationally-costly algorithmic analyses (e.g. Mendes et al 2013; Varela et al 2018a)
Summary
Lateral inhibition is a signalling mechanism that can induce the differentiation of cells in developing tissues (Sternberg 1993; Collier et al 1996). In Collier et al (1996), the authors choose a spatially-discretised model, with dynamics described by systems of differential equations Their analysis highlights in particular that, when the feedback between cells is strong enough, patterns of alternating high and low levels of Notch emerge, that do not depend on specific forms for the regulations of species production, and on the parameters. Boolean lateral inhibition models with one variable per cell, one can use properties of threshold networks (Goles-Chacc et al 1985) to show that all attractors for the asynchronous dynamics are fixed points These stable configurations or patterns that emerge from the simple spatial interaction structure we consider exhibit the same alternation of cells with low and high Notch level observed in the ODE models of Collier et al (1996). We discuss a generalisation of the models and additional open questions in Sects. 5 and 6
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